March 1

Exercise

a) Find all the quadratic residues modulo 19.
b) How many quadratic residues are there modulo 31?
c) Show that the congruence
$(x^2+1)(x^2+2)(x^2-2)\equiv 0 \mod p$
has a solution modulo any prime.

Question

Let
$p=2^n+1$
be a prime (n>0).

Show that a is a quadratic non-residue modulo p if and only if a is a primitive root modulo p.